Difference between revisions of "Metric space"
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*Symmetry: for all <math>x, y \in S</math>, <math>d(x, y) = d(y, x)</math> | *Symmetry: for all <math>x, y \in S</math>, <math>d(x, y) = d(y, x)</math> | ||
*Non-negativity: for all <math>x, y \in S</math>, <math>d(x, y) \geq 0</math> | *Non-negativity: for all <math>x, y \in S</math>, <math>d(x, y) \geq 0</math> | ||
| − | *Uniqueness | + | *Uniqueness: for all <math>x, y \in S</math>, <math>d(x, y) = 0</math> if and only if <math>x = y</math> |
| − | *[[Triangle Inequality]] | + | *The [[Triangle Inequality]]: for all points <math>x, y, z \in S</math>, <math>d(x, y) + d(y, z) \geq d(x, z)</math> |
Intuitively, a metric space is a generalization of the distance between two objects (where "objects" can be anything, including points, functions, graphics, or grades). The above properties follow from our notion of distance. Non-negativity stems from the idea that A cannot be closer to B than B is to itself; Uniqueness results from two objects being identical if and only if they are the same object; and the Triangle Inequality corresponds to the idea that a direct path between points A and B should be at least as short as a roundabout path that visits some point C first. | Intuitively, a metric space is a generalization of the distance between two objects (where "objects" can be anything, including points, functions, graphics, or grades). The above properties follow from our notion of distance. Non-negativity stems from the idea that A cannot be closer to B than B is to itself; Uniqueness results from two objects being identical if and only if they are the same object; and the Triangle Inequality corresponds to the idea that a direct path between points A and B should be at least as short as a roundabout path that visits some point C first. | ||
Revision as of 08:53, 30 November 2006
A metric space is a pair, 
of a set 
and a metric 
. The metric 
represents a distance function between pairs of points of which has the following properties:
- Symmetry: for all
, 
- Non-negativity: for all ,
- Uniqueness: for all ,
if and only if 
- The Triangle Inequality: for all points
, 
, 
if and only if 
, 
Intuitively, a metric space is a generalization of the distance between two objects (where "objects" can be anything, including points, functions, graphics, or grades). The above properties follow from our notion of distance. Non-negativity stems from the idea that A cannot be closer to B than B is to itself; Uniqueness results from two objects being identical if and only if they are the same object; and the Triangle Inequality corresponds to the idea that a direct path between points A and B should be at least as short as a roundabout path that visits some point C first.
Popular metrics
- The Euclidean metric on
, with the "usual" meaning of distance
, with the "usual" meaning of distance- The Discrete metric on any set
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